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• Doubly transitive groups and Hadamard matrices

National University of Ireland, Galway

International workshop on Hadamard matrices and applications RMIT, 29 November 2011

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• Outline

1 The permutation group of a matrix

2 2-Designs, Difference sets, Hadamard matrices

3 Doubly transitive group actions on Hadamard matrices

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• The permutation group of a matrix

Automorphisms of a matrix

Let M be an n × n matrix with entries in a commutative ring R. Then a pair (P,Q) of U(R)-monomial matrices is an automorphism of M if and only if

PMQ−1 = M.

The set of all automorphisms of M forms a group under composition, denoted Aut(M).

But this is not a permutation group...

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• The permutation group of a matrix

Definition Denote by A the set of all entries in M together with 1R. Then the expanded matrix of M is

EM = [ aiajM

] ai ,aj∈A

.

Lemma There exists a homomorphism α : Aut(M)→ Aut(EM), such that the image of (P,Q) ∈ Aut(M) is a pair of permutation matrices.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• The permutation group of a matrix

A permutation quotient

Suppose that M is invertible (possibly over some extension of R). Then P uniquely determines Q:

PMQ−1 = M ⇐⇒ P = MQM−1

So the map β : (P,Q) 7→ P is an isomorphism of groups. Thus we can consider βα(Aut(M)) as a permutation group on the n |A| rows of EM . Linearity of the Aut(M) action gives an obvious system of imprimitivity: blocks are {ari | a ∈ A} . Consider the induced action on this block system. A monomial matrix P can be written in the form XY where X is diagonal and Y is a permutation matrix. The map ρ : P 7→ Y is a homomorphism on any monomial group. This permutation group of degree n is A(M) = ρβ(Aut(M)). Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• The permutation group of a matrix

Cocyclic development

Definition Let G be a finite group and C an abelian group. Then ψ : G×G→ C is a (2-)cocycle if it obeys the identity

ψ(g,h)ψ(gh, k) = ψ(g,hk)ψ(h, k)

for all g,h, k ∈ G.

Definition Let R be a commutative ring, M an n × n matrix R-matrix. Suppose there exist a cocycle ψ : G×G→ U(R) and a set map φ : G→ R such that

M ∼= [ψ(g,h)φ(gh)]g,h∈G .

Then M is cocyclic over G.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• The permutation group of a matrix

Which matrices are cocyclic?

Theorem (de Launey & Flannery)

The matrix M is cocyclic over G if and only if Aut(M) contains a subgroup Γ such that

Γ contains a central subgroup Θ isomorphic to a finite subgroup of U(R). Γ/Θ ∼= G. α(Γ) has induced regular actions on the rows and columns of EM .

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• The permutation group of a matrix

Cocyclic development and A(M)

Suppose that M is cocyclic over G. Then Aut(M) contains a subgroup Γ as in the Theorem. The image of Γ in A(M) is a regular subgroup. So cocyclic development⇒ existence of a regular subgroup in A(M). Unfortunately the converse is not so straightforward: we require a regular subgroup of A(M) to satisfy some additional conditions.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• 2-Designs, Difference sets, Hadamard matrices

Designs

Definition Let V be a set of order v (whose elements are called points), and let B be a set of k -subsets of V (whose elements are called blocks). Then ∆ = (V ,B) is a t-(v , k , λ) design if and only if any t-subset of V occurs in exactly λ blocks.

Definition The design ∆ is symmetric if |V | = |B|.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• 2-Designs, Difference sets, Hadamard matrices

Definition Define a function φ : V × B → {0,1} given by φ(v ,b) = 1 if and only if v ∈ b. An incidence matrix for ∆ is a matrix

M = [φ(v ,b)]v∈V ,b∈B .

Definition The automorphism group of M consists of all pairs of {1}-monomial (i.e. permutation) matrices such that

PMQ> = M.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• 2-Designs, Difference sets, Hadamard matrices

Definition An automorphism of the design ∆ is a permutation σ ∈ Sym(V ) which preserves B setwise.

An automorphism σ of ∆ induces a permutation of the rows of M. In fact, Aut(∆) = A(M). It is known that for symmetric 2-designs

Aut(∆) ∼= Aut(M) ∼= A(M).

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• 2-Designs, Difference sets, Hadamard matrices

Difference sets

Let G be a group of order v , and D a k -subset of G. Suppose that every non-identity element of G has λ representations of the form did−1j where di ,dj ∈ D. Then D is a (v , k , λ)-difference set in G. e.g. {1,2,4} in Z7.

Theorem If G contains a (v , k , λ)-difference set then there exists a symmetric 2-(v , k , λ) design on which G acts regularly. Conversely, a 2-(v , k , λ) design on which G acts regularly corresponds to a (v , k , λ)-difference set in G.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• 2-Designs, Difference sets, Hadamard matrices

Definition An automorphism of a Hadamard matrix H is a pair of {±1}-monomial matrices such that

PHQ> = H.

The set of all automorphisms form a group, Aut(H).

A(H) is a permutation group on the rows of H. The kernel of the map Aut(H)→ A(H) consists of automorphisms whose first component is diagonal. (−I,−I) is always an automorphism of H, so that this kernel if always non-trivial. If H is cocyclic, then A(H) contains a regular subgroup.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• 2-Designs, Difference sets, Hadamard matrices

Hadamard matrices, 2-designs and difference sets

Lemma There exists a Hadamard matrix H of order 4t if and only if there exists a 2-(4t − 1,2t − 1, t − 1) design D. Furthermore Aut(D) embeds into the stabiliser of a point in A(H).

Corollary

Suppose that H is developed from a (4t − 1,2t − 1, t − 1)-difference set. Then the stabiliser of the first row of H in A(H), is transitive on the remaining rows of H.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• 2-Designs, Difference sets, Hadamard matrices

Example: the Paley construction

The existence of a (4t − 1,2t − 1, t − 1)-difference set implies the existence of a Hadamard matrix H of order 4t .

Let Fq be the finite field of size q, q = 4t − 1. The quadratic residues in Fq form a difference set in (Fq,+) with parameters (4t − 1,2t − 1, t − 1), (Paley).

Let χ be the quadratic character of of F∗q, given by χ : x 7→ x q−1

2 , and let Q = [χ(x − y)]x ,y∈Fq . Then

H =

( 1 1

1 >

Q − I

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• Doubly transitive group actions on Hadamard matrices

Doubly transitive group actions on Hadamard matrices

Two constructions of Hadamard matrices: from (4t − 1,2t − 1, t − 1) difference sets, and from (orthogonal) cocycles.

Problem How do these constructions interact? Can a Hadamard matrix support both structures? If so, can we classify such matrices?

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• Doubly transitive group actions on Hadamard matrices

Motivation

Horadam: Are the Hadamard matrices developed from twin prime power difference sets cocyclic? (Problem 39 of Hadamard matrices and their applications) Jungnickel: Classify the skew Hadamard difference sets. (Open Problem 13 of the survey Difference sets). Ito and Leon: There exists a Hadamard matrix of order 36 on which Sp6(2) acts. Are there others?

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011

• Doubly transitive group actions on Hadamard matrices

Doubly transitive group actions on Hadamard matrices

Lemma Let H be a Hadamard matrix developed from a (4t − 1,2t − 1, t − 1)-difference set, D in the group G. Then the stabiliser of the first row of H in A(H) contains a regular subgroup isomorphic to G.

Lemma Suppose that H is a cocyclic Hadamard matrix with cocycle ψ : G×G→ 〈−1〉. Then A(H) contains a regular subgroup isomorphic to G.

Corollary If H is a cocyclic Hadamard matrix which is also developed from a difference set, then A(H) is a doubly transitive permutation group.

Padraig Ó Catháin Doubly transitive groups and Hadamard matrices 4 November 2011